Understanding Quantum Evolution Complexity: New Insights from Recent Research

Recent research by Carlo Cafaro, Leonardo Rossetti, and Paul M. Alsing, titled "Complexity of Quantum-Mechanical Evolutions from Probability Amplitudes," investigates the complexities involved in quantum Hamiltonian evolutions. The study, submitted on August 26, 2024, focuses on both time-optimal and sub-optimal evolutions connecting arbitrary source and target states on the Bloch sphere, utilizing the Fubini-Study metric.

The authors describe unitary Schrödinger quantum evolution through various metrics, including path length, geodesic efficiency, speed efficiency, and curvature coefficients. They transition from a classical probabilistic framework to a deterministic quantum setting, proposing a definition of quantum evolution complexity and introducing a quantum complexity length scale.

A key finding is that efficient quantum evolutions tend to exhibit lower complexity than inefficient ones. However, the research also notes that complexity is not solely determined by path length; longer paths with significant curvature can demonstrate lower complexity than shorter, less curved paths. This nuanced understanding of quantum evolution could have implications for optimizing quantum computing processes and enhancing the efficiency of quantum algorithms.

The full paper can be accessed via arXiv at arXiv:2408.14241.